Instructor: |
Nils Bruin
nbruin@sfu.ca
SC K 10507
(778) 782 3794
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Webpage: |
http://www.cecm.sfu.ca/~nbruin/math845ADS |
Textbook:
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Silverman, Joseph H. The arithmetic of dynamical systems.
Graduate Texts in Mathematics, 241. Springer-Verlag, New York, 2007.
x+511 pp. ISBN: 978-0-387-69903-5.
Questions and errata
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Course materials: |
Assignments and handouts are available on this
page. |
Prerequisites:
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- Algebraic number theory (MATH 842). Can be taken together with this course.
- Algebraic geometry is useful but not required.
- A reasonable knowledge of algebra (in particular galois theory)
will be very useful.
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Meetings: |
Scheduled timeslot is Monday, Wednesday 9:30 - 11:20 in K9509. However, if that is not suitable for a participant,
we can try to change to a slot that DOES work for everybody. See
this table.
First meeting will be to determine our further schedule and will be Wednesday January 6.
Lecture schedule
Jan 11. Nils
Chapter 1: Introduction to classical discrete dynamical systems
Critical points, fixed points, multipliers, relations on multipliers.
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Jan 13. Fatou and Julia sets: Pictures.
Start on p-adics.
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Jan 18. Kevin
Chapter 2: p-adic dynamical systems with good reduction (part 1)
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Jan 20. Local rings and their completions, projective limits. Resultants as linear algebra.
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Jan 25. Karen
Chapter 2: p-adic dynamical systems with good reduction (part 2)
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Jan 27. Introduction to heights and geometry (roughly sections 3.1, 3.2) |
Feb 1. Kate
Chapter 3: Canonical heights
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Feb 3. Heights and morphisms |
Feb 8. Kate
Chapter 3: Dynamical Diophantine Approximation
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Feb 10. Iain
Newton Polygons
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Feb 15 - 26: Olympic break |
Mar 1 Alex Molnar
Chapter 5: Nonarchimedean Julia and Fatou sets; the dynamics of (z^2-z)/p
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Mar 3 Nils Bruin
Proof of Proposition 5.24 (step to symbolic dynamical characterisation of the Julia set of (z^2-z)/p, Theorem 5.26 (towards a non-archimedean Montel Theorem).
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Mar 8 Alex Molnar
Julia sets and the closure of the periodic points.
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Mar 10. Exercises |
Mar 13 - 17: Arizona Winter School |
Mar 22
Sander Dahmen
Dynatomic polynomials (Section 4.1)
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Mar 24. Exercises |
Mar 29
Sander Dahmen
Dynatomic Curves
Nils Bruin
Moduli spaces of rational function; Multiplier spectra
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Mar 31
Nils Bruin
Dynamics from multiplicative groups; Twists
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Apr 5: Easter |
Apr 7
Nils Bruin
Chebyshev Polynomials
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Apr 12
Nils Bruin
Elliptic curves; Lattès maps
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Apr 14
Nils Bruin
Multipliers for flexible Lattès maps
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Description: |
Arithmetic Dynamics is a relatively new area of research. It may be
viewed as the transposition of classical results in the theory of
Diophantine equations to the setting of discrete dynamical systems,
such as iterated polynomial maps.
The interplay between arithmetic and dynamics yields a particularly
rich structure, which resembles the much older and established field
of arithmetic algebraic geometry. The course text gives a particularly
accessible introduction to the field.
We will follow the book quite closely. The chapters are:
- Classical dynamics
- Dynamics over local fields: good reduction
- Dynamics over global fields
- Families of dynamical systems
- Dynamics over local fields: bad reduction
- Dynamics associated to algebraic groups
- Dynamics in dimension greater than one
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Format: |
Given the advanced nature of the material, the course will be run in a
mixed seminar/workgroup style. Every week, there will be 2 hours of
seminar and an additional 2 hours of workgroup, where the lecturer and
the students can look at problems and further delve into the theory.
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Grading: |
- Participating students are expected to prepare at least one seminar
contribution, which will be assessed and count towards the grade.
- Students will regularly hand in assignments, which will be marked.
- Students will be graded on participation in the problem sessions.
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Lecture schedule: |
To be announced |
Field trip: |
The Arizona Winter School 2010, held March 13-17, 2010 at the University of Arizona in Tucson, AZ,
will be on Number Theory and Dynamics. It is warmly recommended that you attend this event. Application deadline for students seeking funding is
January 11, 2010. |